The complex activity of neural populations in the Prefrontal Cortex (PFC) is a hallmark of high-order cognitive processes. How these rich cortical dynamics emerge and give rise to neural computations is largely unknown. Here, we infer models of neural population dynamics that explain how PFC circuits of monkeys may select and integrate relevant sensory inputs during context-dependent perceptual decisions. A class of models implementing linear dynamics accurately captured the rich features of the recorded PFC responses.These models fitted the neural activity nearly as well as a factorization of population responses that had the flexibility to capture non-linear temporal patterns, suggesting that linear dynamics is sufficient to recapitulate the complex PFC responses in each context. Two distinct mechanisms of input selection and integration were consistent with the PFC data. One mechanism implemented recurrent dynamics that differed between contexts, the other a subtle modulation of the inputs across contexts. The two mechanisms made different predictions about the contribution of non-normal recurrent dynamics in transiently amplifying and selectively integrating the inputs. In both mechanisms the inputs were inferred directly from the data and spanned multi-dimensional input subspaces. Input integration likewise consistently involved high dimensional dynamics that unfolded in two distinct phases, corresponding to integration on fast and slow time-scales. Our study offers a principled framework to link the activity of neural populations to computation and to find mechanistic descriptions of neural processes that are consistent with the rich dynamics implemented by neural circuits.